research article a betweenness calibration topology...

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2013 Article ID 212074 9 pageshttpdxdoiorg1011552013212074

Research ArticleA Betweenness Calibration Topology Optimal Control Algorithmfor Wireless Sensor Networks

Ting Yang Zhixian Lin and Bo Yuan

School of Electrical Engineering and Automation Tianjin University Tianjin 300072 China

Correspondence should be addressed to Ting Yang yangtingtjueducn

Received 24 July 2013 Accepted 19 August 2013

Academic Editor Zhaoxia Wang

Copyright copy 2013 Ting Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In self-organized wireless sensor networks (WSNs) any two sensor nodes can connect if they are placed in each otherrsquoscommunication range Therefore the physical topology of WSNs is usually a strongly connected topology Sensor nodes shouldfrequently receive and process data from their large number of neighbors which will consume great amounts of energy Shockingwireless channel collision also causes low throughput and high loss packets ratio during data transmission To improve thetransmission performance and save scarce energy a logical topology generating from the physical one is necessary for the self-organizedWSNs Based on the complex network theory this paper proposed a novel betweenness addition edges expansion algorithm(BAEE)With betweenness calibration BAEE algorithm expanded theminimum-cost edges to optimize the network topology Twoperformance metrics-connectivity functions robustness function 119877(119866) and efficiency function 119864(119866) were utilized to evaluate thenetwork capability of the robustness and invulnerability 119877(119866) is the parameter to measure the topology connectivity and 119864(119866) isthe parameter to evaluate the network exchanging information capability Based on the simulation under various random failuresand intentional attack scenarios BAEE can effectively optimizeWSNsrsquo topology and improve the networkrsquos robust connectivity andextremely efficient exchanging information capability

1 Introduction

Wireless sensor networks (WSNs) are a class of self-organizedwireless communication networks in which many sensornodes collect process and exchange information acquiredfrom the physical environment or the monitor objects andthen send it to the external base station called Sink [1]WSNshave awide range of potential applications including environ-ment monitoring smart grid medical systems and roboticexploration [2]

There are two main difficulties in WSNsrsquo design (1)the limited and nonreplenishable energy supply and (2)the limited transmission bandwidth and high packet lossrate caused by the out-of-order distributed communicationHence the energy control algorithm and robust infrastruc-ture are necessary to prolong the networksrsquo lifetime andimprove the communication performance

Topology optimal control (TOC) is to design a good logi-cal network infrastructure one of the key techniques used inwireless self-organized sensor networks [3] In a network ifthere is at least one route to connect any two sensor nodes

such network is regarded as a connected one Because of theomnidirectional antenna any two nodes in WSNs can com-municate if the Euclidean distance between them is less thanthe communication rangeTherefore the physical topology ofWSNs is usually a strongly connected topology Any nodewillfrequently receive and process data from the quantity of itsneighbors which will consume great amounts of energy Theminimum energy network connectivity (MENC) problemwas defined and proved to be an NP-complete problem [4]

In the research on TOC the previous research can beclassified into two types based on the optimized objects phys-ical topology control algorithms (PTCA) and logical topologycontrol algorithms (LTCA) [5 6] PTCA adjust sensor nodesrsquotransmission power to control the physical topology On theother hand LTCA restrict one node connected with a certainnumber of neighbors to satisfy the network connectivityThisneighbor reduction mechanism helps to reduce the routingoverhead and relieve the channel collision problems

Different from the wired communication network suchas IP network WSN is one type of dynamic networks

2 International Journal of Distributed Sensor Networks

There are many factors causing the dynamic structuremdashfrom system hardware to applicationmdashfor unattended sensornodes with miniature sizes (mm scale for smart dust motes)limited battery-power and low reliable hardware circuitswhen coping with harsh conditions Other factors thatmay affect network connectivity and communication amongsensor nodes are fading signal strength obstacles weatherconditions interference and so forth [7 8] An immutabletopology structure is not enough for the WSNs and anydynamic change will break original optimization and reducethe network performance

To overcome this critical problem this paper proposeda novel betweenness addition edges expansion algorithm(BAEE) With the betweenness parameter BAEE algorithmexpanded the minimum-cost edges to optimize the networktopology with maximum improving of the efficiency func-tion values The preliminary simulation results comparedwith Fiedler-vector-based strategy showed that our algorithmcould obtain more robust topology with higher invulnera-bility under both the random failures and intentional attackscenarios

This paper is organized as follows Section 1 introducesthe TOC problem in WSNs and Section 2 presents therelated work The problemrsquos mathematic description andmodel building are presented in Section 3 Section 4 proposesthe BAEE algorithm in detail Section 5 presents simulationresults to demonstrate the effectivity of the algorithm Section6 concludes the paper

2 Related Work

There are three types of approaches in the previous TOCresearch presented as follows (1) Control each nodersquos emis-sion power to reduce the strong connectivity of the physicaltopology and to effectively save the energy consumption andprolong network lifespan Rodoplu andMeng [9] introducedthe notion of relay region and enclosure for the purpose ofpower control It was shown that the network was stronglyconnected if every node maintained links with the nodesin its enclosure With reducing the transmission power thetopology connectivity becomes thin Building a minimum-power-connected topology is a multiobjective optimizationproblem (2) Reduce the total number of working nodesin WSNs and let other nodes suspend to hibernate It canalso reduce the topology complexity Moreover the approachhelps to reduce the interference that exists in wireless net-work which means that a greater signal-to-noise ratio willbe obtained at receiving nodes The most common schemesbased on this principle are sensor-MAC (S-MAC) [10]timeout-MAC (T-MAC) [11] and data-gathering MAC (D-MAC) [12] (3) Control sensor nodersquos logical degree in its log-ical topology thus helping to reduce MAC layer contentionand improve space reuse A less nodersquos logical degreemay alsohelp to mitigate the hidden and exposed terminal problemsClustering topology control strategy is one of the effectiveapproaches similar to spanning-tree structure in WSN Thelow-energy adaptive clustering hierarchy (LEACH) [13] isthe most notable clustering algorithm for wireless sensornetworks LEACH combines the ideas of energy-efficient

cluster with application-specific data aggregation to achievegood performance Its improved algorithm power-efficientgathering in sensor information systems (PEGASIS) [14] isa chain-based clustering scheme Another effective topologystructure is the spanning-tree [15] Li et al [16] proposeda fully distributed topology control algorithm called LMSTA similar method 119896-local MST was addressed by Li et al[17]

With the number of sensor nodes increasing the topol-ogy of large-scale WSNs becomes more and more complexand TOC as a type of multiobjective optimization problemsis very difficult to explore the global optimal solution suchas the degree-constrained minimum spanning-tree problemSome heuristic methods were developed to improve the opti-mization performance [18ndash23] In [18] the authors proposedtwo heuristics based on a minimum spanning-tree algorithmand a broadcast incremental power method respectivelyKonstantinidis developed a genetic algorithm with localsearch that performs better than the MST heuristic [19] Guopresented an improved discrete particle swarm optimizationalgorithm for generating topology schemes [20] A simulatedannealing algorithm was designed in [21] and it is alsoapplied to solve the problem of minimizing broadcast treeone type of the physical topology control problems [22] In[23] ant colony optimization a framework inspired by the antforaging behavior in the area of swarm intelligence is appliedto physical topology control

The above heuristic algorithms focused on the solutionprocedure of optimization problem itself in which topologycontrol had been abstracted into the multiobjective opti-mization problem On the different view to analyze thetopology control problem we use the complex networktheory to calculate the networkrsquos long-range and short-rangeconnectivity and then a novel BAEE algorithm is proposedto improve the networksrsquo robustness and invulnerability withthe minimum-cost edges expanded

3 The Network Model and the Parameters ofComplex Network

The formal definition of the TOC problem in WSNs ispresented as follows In a special sensor area there are a setof 119899 wireless nodes 119881 = V

1 V2 V

119899 119864 = 119890

1 1198902 119890

119899

is the set of communication links When an adjacent pair V119895

V119896shows the same wireless medium 119890

119894(V119895 V119896) indicates that

both nodes are within their wireless transmitting ranges 1205820

that is 119864 = 119890119894(V119895 V119896) | 119863(V

119895 V119896) le 1205820 V119895 V119896isin 119881 Therefore

the wireless sensor network is represented as a simple digraph119866 = (119881 119864) Because of the omnidirectional antenna inWSNsany two nodes can communicate if the Euclidean distancebetween them is less than the communication range There-foreWSNsrsquo topology is usually strongly connectedThe com-plex network theory is utilized to analyze this type of stronglyconnected topology in this paper Some parameters of com-plex network are presented firstly in the following section

31 The Parameters of Complex Network A complex net-workrsquos attribute can be described by its key parameters degree

International Journal of Distributed Sensor Networks 3

distribution clustering coefficient average path length andbetweenness [24 25]

(1) Cumulative Degree Distribution Function The degree ofa node in a network is the number of connections and thedegree distribution 119901(119896

1015840) is the probability distribution of

these degrees over the whole networkThe cumulative degreedistribution function 119875

119888(119896) is the probability distribution of

all of the nodes whose degree is not less than 119896 Consider thefollowing

119875119888(119896) =

119896max

sum

1198961015840ge119896

119901 (1198961015840) (1)

(2) Clustering Coefficient In graph theory a clustering coef-ficient is a measure of the degree to which nodes in a graphtend to cluster together Firstly the local clustering coefficient119862119894of a node V

119894in a graph quantifies how close its neighbors

are to being a clique that is complete graph Let 120582119866(V) be

the number of triangles on V isin 119881(119866) for undirected graph119866 That is 120582

119866(V) is the number of subgraphs of 119866 with three

edges and three nodes one of which is V Let 119879119866(V) be the

number of triples on V isin 119881(119866) That is 119879119866(V) is the number

of subgraphs (not necessarily induced) with two edges andthree nodes one of which is V such that V is incident to bothedges Then local clustering coefficient 119862

119894can be defined as

119862119894=

120582119866(V)

119879119866(V)

(2)

The clustering coefficient for the whole network is givenas the average of the local clustering coefficients of all of thenodes 119873 as follows

119862 =1

119873

119873

sum

119894=1

119862119894 (3)

Evidence suggests that in most real-world networksnodes tend to create tightly knit groups characterized bya relatively high density of ties this likelihood tends tobe greater than the average probability of a tie randomlyestablished between two nodes

(3) Average Path Length Average path length 119871 is definedas the average number of steps along the shortest paths forall possible pairs of network nodes It is a measure of theefficiency of information ormass transport on a networkThedefinition is shown as

119871 =1

119873 (119873 minus 1) 2sum

1le119894119895le119873

119889119894119895 (4)

Average path length is one of themost robust measures ofnetwork topology

(4) Betweenness There are two definitions the vertexbetweenness 119861(V) and the edge betweenness 119861(119890) Here weused the 119861(V) as the example Betweenness 119861(V) a centrality

measure of a node within a graph centrality quantifies thenumber of times a node acts as a bridge along the shortestpath between two other nodes Consider the following

119861 (V) = sum

V = 119894 V = 119895 119894 = 119895

120590119894119895(V)

120590119894119895

(5)

Here 120590119894119895is the total number of the shortest paths from

node 119894 to node 119895 and 120590119894119895(V) is the number of those paths that

pass through V

32 Two Evaluation Functions for Measuring the NetworkrsquosRobustness and Invulnerability Based on the above fourparameters connectivity robustness function and efficiencyfunction can be defined and utilized to evaluate the networkrsquosrobustness and invulnerability

(1) Connectivity Robustness Function 119877(119866) Connectivityrobustness refers to maintaining the connection capability ofthe remaining nodes when some of the nodes or edges in thenetwork were removed [26] In the network 119866 = (119881 119864) with119899 nodes the connectivity robustness is defined as follows

119877 (119866) =

119878 (1198661015840)

119873 minus 119873119903

(6)

Here 119878(1198661015840) is the largest connected component remain-ing after the removal of 119873

119903nodes The connectivity robust-

ness is normalized that is 0 lt 119877(119866) le 1Themaximumvalueof the connectivity robustness function 119877(119866) = 1 is obtainedin the case that 1198661015840 is also a connected graph after 119873

119903nodes

were removed

(2) Efficiency Function 119864(119866) Instead of 119871 and119862 the networkis characterized in terms of how efficiently it propagatesinformation on a global and on a local scale respectivelydefined as the global efficiency function 119864glob(119866) and localefficiency function 119864loc(119866) [25 27] We assume that theefficiency 120576

119894119895in the communication between nodes 119894 and 119895 is

inversely proportional to the shortest distance 120576119894119895

= 1119889119894119895

forall119894 119895 In this definition when there is no path in the graphbetween 119894 and 119895119889

119894119895= +infin and consistently 120576

119894119895= 0The global

efficiency of the graph 119866 can be defined as

119864glob (119866) =

sum119894 = 119895isin119866

120576119894119895

119873(119873 minus 1)=

1

119873 (119873 minus 1)sum

119894 = 119895isin119866

1

119889119894119895

(7)

The local efficiency function can be defined as the averageefficiency of local subgraphs as follows

119864loc (119866) =1

119873sum

119894isin119866

119864 (119866119894)

119864loc (119866) =1

119896119894(119896119894minus 1)

sum

119897 =119898isin119866

1

1198891015840

119897119898

(8)

where 119866119894is the subgraph of the neighbors of 119894 which is made

by 119896119894nodes and at most 119896

119894(119896119894minus 1)2 edges It is important


to notice that the quantities 1198891015840

119897119898 are the shortest distances

between nodes 119897 and 119898 calculated on the graph 119866119894

Both the global and the local efficiency are alreadynormalized that is 0 le 119864glob(119866) le 1 and 0 le 119864loc(119866) le 1The maximum values of the efficiency 119864glob(119866) = 1 and119864loc(119866) = 1 are obtained in the ideal case of a completelyconnected graph that is in the case in which the graph119866 hasall of the 119873(119873 minus 1)2 possible edges and 119889

119894119895= 1 forall119894 119895

In the efficiency-based formalism a network is extremelyefficient in exchanging information both on a high global andlocal efficiency functions value Moreover the descriptionof a network in terms of its efficiency can be extended tounconnected networks and more important with only a fewmodifications to weighted networks A weighted network isa case in which there is a weight associated with each of theedges Such a network needs two matrices to be describedConsider the following

(1) The usual adjacency matrix [119886119894119895] telling about the

existence or nonexistence of a link is defined asfollows

119886119894119895

= 1 119890119894119895

isin 119864

0 otherwise(9)

(2) The second weights matrix associated with each link[119908119894119895] where 119908

119894119895can be defined as communication

cost depended on the optimal problem Obviouslyweighted network optimal problem such as TSP(traveling salesman problem) is more complex thantopology control

In this paper we focus instead on the simpler case ofunweighted networks topology We will use the connectivityrobustness function and the global efficiency function toevaluate the networkrsquos robustness and invulnerability underthe random failures and the intentional attacks

4 Description of Betweenness Addition EdgesExpansion Algorithm

In order to improve the networkrsquos robustness and invulnera-bility under the random failures and the intentional attacksa novel betweenness addition edges expansion algorithm isproposed Based on the traffic analysis in practical WSNswe find that the communication connection is establishedusually by events driven in which both the vertex between-ness 119861vet and edge betweenness 119861edg follow heavy-taileddistribution As in the above definition the betweenness isthe number of the shortest paths through node V

119894or edge

119890119894 which shows the importance of V

119894and 119890

119894in network

transmission The vertex V119894with high-betweenness 119861vet bears

more packets switching which is the core vertex in thenetwork The edge 119890

119894with high-betweenness 119861edg bears more

traffic flows which is the key edge for the networkrsquos con-nectivity In order to improve the networks robustness andavoid transmitting congestion networkrsquos core parts shouldbe identified BAEE carefully selects special vertex parts andconnects edges with betweenness addition strategy After the

optimal operation the network diameter can be effectivelyreduced and the transmission delay will be shortened BAEEprocess is given as follows

(1) According to the vertex betweenness 119861(V) formulaand network adjacency matrix [119860]

119899times119899 each vertex

betweenness is calculated and saved in the columnvector 119861

119899 Consider the following

119861119899= [119887 (V

1) 119887 (V

2) 119887 (V

119899)]119879

(10)

(2) Using the vector betweenness column vector 119861119899 a

new betweenness plus column vector [119861+

]119898is calcu-

lated where 119898 = 1198622

119899 as follows

119861+

(119896) = 119887 (V119894) + 119887 (V

119895)

119896 = (119894 minus 1) times 119899 + 119895

(11)

(3) Sort [119861+]119898in descending order and generate [119861

+

]1015840

119898

Here we used the bubble method to sort columnvector [119861+]

119898 which has lower space complexity119874(1)

and better stability compared with other sortingalgorithmsThen the elements in front of [119861+]

1015840

119898have

larger value of betweenness addition

The process of the bubble method is shown asfollows(a) Compare the first element that is 0th with

the latter element if smaller then switchSequentially compare119898 times element andeventually change the smallest value ele-ment into 119898th unit the element 119887(119898) doesnot move anymore

(b) Repeat step (a) and sequentially compareuntil the (119898 minus 1)th element Eventually theminimum value in the front of (119898 minus 1)

elements is moved to the (119898 minus 1)th unit119887(119898 minus 1) does not move

(m) Compare 119887(0) and 119887(1) if 119887(0) lt 119887(1)switch each other Then 119887(0) is maximumand the array is in descending order Thebubble method terminates

(4) Check [119861+

]1015840

119898from the first elements 119887

+(0) For

119861+

(119896) = 119887(V119894) + 119887(V

119895) if vertex parts V

119894and V

119895have

connection edge that is 119890(V119894 V119895) isin 119864 then check the

next element of [119861+]1015840

119898

(5) Else if 119890(V119894 V119895) notin 119864 then add an edge between

vertex parts V119894and V119895Then calculate the connectivity

robustness function 119877(119866) and efficiency function119864(119866)

(6) If both119877(119866) and119864(119866) reach the optimization require-ments the algorithm terminates Otherwise return tostep (4) to look for other connected edges

BAEE algorithm is presented in Algorithm 1


According to the above optimization approach the exper-imental simulation was taken to evaluate the algorithmrsquosperformance The detailed analysis about results was shownin the following section

5 Simulation and Performance Evaluation

The simulation scenario is that 100 sensor nodes wererandomly placed in a 900m times 900m field Each nodersquos radiopropagation range is 300m After the self-organized processa strongly connected physical topology is established Toreduce the interference the neighbors of each sensor node arecontrol based on the traffic requirements and then a logicaltopology is generated which is the topology that we reallyneed for data transmission shown in Figure 1

The connectivity robustness function 119877(119866) and efficiencyfunction119864(119866) for the initial network are calculated as follows119877(119866) = 1 because it is a connected graph 119864(119866) = 0226 Inthe simulation BAEE algorithm is used to optimize the net-work topology compared with Fiedler-vector-based strategy(FVBS) another well-known method for TOC presented in[28] FVBS main idea is adding a link between a node pairwith the maximal |119906

119894minus 119906119895| the absolute difference between

the 119894th and 119895th elements of the Fiedler vector of 119866 Becausethe Fiedler vector is related to the algebraic connectivity of119866to maintain the fairness of evaluation the simulation resultsare analyzed through the connectivity robustness function119877(119866) and efficiency function 119864(119866) except for the algebraicconnectivity

In the simulation firstly using BAEE and FVBS tooptimize the original topology the new topologies1198661015840BAEE and1198661015840

FVBS are generated Tomaintain the fairness the same num-ber of edges is added in 119866

1015840

BAEE and 1198661015840

FVBS shown in Table 1Then the identical random failures and the intentional

attacks are applied on the two 1198661015840 The robustness and

invulnerability are evaluated by the two performance metrics119877(119866) and 119864(119866)

51 The Experiments under Random Failures Random fail-ures mean that nodes in the network are randomly failedand at the same time the edges connecting with the failurenodes are also failed Because of the low reliable hardwarecircuits the limited battery-power and the harsh wildernessconditions the case of sensor node failed often occurs inthe practical application Figure 2 shows the connectivityrobustness function value for the increasing of the numberof random failure nodes From Figure 2 we can observethat the two optimized topologies have higher 119877 value thanthe original network confronting random failures MoreoverBAEE is better than FVBS the 119877 value has an average523 increase which means that the optimized network hasthe stronger capability of maintaining connectivity Differentfrom the other two curves the 119877(119866

1015840

BAEE) curve of BAEEis stable For example at 11 failure nodes scenario the119877(1198661015840

BAEE) curve does not shake different from the sharpdecline of 119877(119866

1015840

FVBS) and 119877(119866) curves which indicates thatBAEE algorithm has better ldquoresistancerdquo

Figure 3 shows the efficiency function 119864(119866) under ran-dom failures As the number of the failure nodes increasesthe efficiency function of the three networks decreases Thereason is that failure nodes make certain shortest paths bro-ken But 119864(119866

1015840

BAEE) is higher than 119864(1198661015840

FVBS) and 119864(119866) whoseincrease rates are 1378 and 2359 respectively This isextreme efficiency showed that BAEE algorithm can optimizenetwork and reach extreme efficiency in exchanging informa-tion for ubiquitous data-centric wireless sensor networks

52 The Experiments under Intentional Attacks Intentionalattack is another kind of accident for wireless sensor net-works Based on partial information of network enemy canaccurately attack the weakest parts and break down the wholesystem So a network should have more robust topologyto resist intentional attacks In the following experimentstwo types of attacks are simulated (1) make nodes withhigh vertex betweenness fail (2) make links with high edgebetweenness fail The two metrics 119877(119866) and 119864(119866) are alsoused to measure the algorithmsrsquo performance

Figure 4 presents the connectivity robustness function119877(119866) under the intentional attacks with high-betweennessnodes failed From the three curves we can find that bothBAEE and FVBS algorithms improve the original networkrsquosinvulnerability 1198661015840BAEE is also stronger than 119866

1015840

FVBS when thenumber of failed nodes is more than 6 The gap is 2574approximately When the number of failed nodes is continu-ally increasing and more than 13 the values of 119877(119866

1015840

BAEE) and119877(1198661015840

FVBS) have a sharp decline and coincide with 119877(119866) Thereason is that the original network has its inherent structurequality and TOC algorithms can just improve the networkperformance limited

The efficiency function 119864(119866) against nodesrsquo failure isshown in Figure 5 BAEE algorithm optimized the networkand reached a high value of 119864(119866

1015840

BAEE) The average is higherthan 119864(119866

1015840

FVBS) 2298 Moreover we found that the sametwo aberration points occurring in above experiments alsoappear in this experiment when the number of failed nodesismore than 6119864(119866

1015840

FVBS) and119864(119866) curves present a dump butthe network topology optimized by BAEE algorithm escapesthis shake showing stronger stability When the number offailed nodes is more than 13 both 119864(119866

1015840

BAEE) and 119864(1198661015840

FVBS)have a sharp decline and coincide with 119864(119866) proving thenetworkrsquos inherent structure quality

Another attack strategy high-betweenness links failedis implemented in the experiments The top 20 high-betweenness links are sequentially broken to evaluate thenetworkrsquos performance the curves of 119877(119866) are presented inFigure 6 The results show that under the high-betweennessedgesrsquo attack FVBS algorithm cannot improve the networkrsquosinvulnerability capability anymore shown as the two curves119877(1198661015840

FVBS) and 119877(119866) coinciding However BAEE algorithm iseffective under this kind of attack While the broken links areless than 12119877(119866

1015840

BAEE) is higher than119877(1198661015840

FVBS) and119877(119866) 989averagely

Figure 7 presents the efficiency function 119864(119866) under theedges intentional attack BAEE also reaches a higher119864(119866

1015840

BAEE)

value than FVBS 119864(1198661015840

FVBS) and original network 119864(119866) in


Betweenness Addition Edges Expansion algorithm

(1) Calculate each vertexrsquos betweenness with the vertex betweenness 119861(V) formula

119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895

(2) Save each vertex betweenness 119861(V) in the column vector 119861119899= [119887(V

1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)

(6) End for(7) End for(8) For 119894 = 0 to 119899 lowast (119899 minus 1)2

(9) For 119895 = 119894 + 1 to 119899 lowast (119899 minus 1)2

(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895


(15) If no edge connected vertex parts V119894and V

119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)

(16) 119886[119894][119895] = 119886[119895][119894] = 1 here 119886[][] is the element of adjacency matrix [119860]119899times119899

(17) End if(18) If 119877(119866) gt 119877(119866)req ampamp 119864(119866) gt 119864(119866)req(19) Break(20) End for

Algorithm 1 Pseudocode of BAEE algorithm

Table 1 Added edges in 1198661015840

BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)

Figure 1 Original logical topology of WSN

which an average increasing rate is 3141 for 119864(1198661015840

FVBS) and5088 for 119864(119866) These results indicate that BAEE algorithmhas obvious advantages against edges intentional attacks

6 Conclusions

Because of the omnidirectional antenna in WSNs any twosensor nodes can connect if they are placed in each otherrsquoscommunication range Therefore the physical topology ofWSNs is usually a strongly connected topology Anyoneshould frequently receive and process data from the quantityof its neighbors which will consume large amounts of

0010203040506070809

1

0 5 10 15 20 25 30 35 40 45 50Number of failed nodes

Original topologyBAEEFVBS

Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)

Figure 2 Connectivity robustness function versus the number offailed nodes under random failures

energy Shocking wireless channel collision also causes lowthroughput and high loss packets ratio in data transmissionTo improve theWSNs transmission efficiency and save scarce


0

005

01

015

02

025

03


Effici

ency

func

tionE(G)


Figure 3 Efficiency function versus the number of failed nodesunder random failures

0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


Figure 4 Connectivity robustness function versus the number offailed nodes under intentional attack

energy a logical topology generating from a physical oneand further dynamic optimization are necessary for the self-organized wireless sensor networks

With topology vulnerability analysis this paper proposesone topology optimization control algorithmmdashBAEE Thealgorithm calculates the vertex betweenness and expandedspecial edges with the minimum cost Two metrics the con-nectivity robustness function 119877(119866) and efficiency function119864(119866) are utilized to measure the network performance 119877(119866)

is the metric to measure topology connectivity and 119864(119866)

is the metric to evaluate the network exchanging informa-tion capability Detailed definitions are presented in thispaper Using numerical experimental simulations under var-ious random failures and intentional attack scenarios wemeasured the performance of BAEE and compared it withthe Fiedler-vector-based strategy in TOC Results were very

0

005

01

015

02

025

03


Effici

ency

func

tionE(G)


Figure 5 Efficiency function versus the number of failed nodesunder intentional attack

07

075

08

085

09

095

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Number of failed links


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)

Figure 6 Connectivity robustness function versus the number offailed links under intentional attack

01012014016018

02022024026

0 1 2 3 4 5 6 7 8 9 1011121314151617181920Number of failed links

Effici

ency

func

tionE(G)


Figure 7 Efficiency function versus the number of failed linksunder intentional attack


promising and showed that our novel algorithmrsquos perfor-mance is much better than others in reaching high connec-tivity robustness function value and efficiency function valuewhichmeans that the optimized network by BAEE has robustconnectivity and extremely efficient exchanging informationcapability

Acknowledgments

This work was sponsored by the National Natural ScienceFoundation of China no 61172014 the Natural Science Foun-dation of Tianjin no 12JCZDJC21300 and the National Pro-gram of International SampT Cooperation no 2013DFA11040

References

[1] R V Kulkarni A Forster and G K Venayagamoorthy ldquoCom-putational intelligence in wireless sensor networks a surveyrdquoIEEE Communications Surveys and Tutorials vol 13 no 1 pp68ndash96 2011

[2] A Alamri W S Ansari M M Hassan et al ldquoA survey onsensor-cloud architecture applications and approachesrdquo Inter-national Journal of Distributed Sensor Networks vol 2013Article ID 917923 18 pages 2013

[3] N Ababneh ldquoPerformance evaluation of a topology controlalgorithm for wireless sensor networksrdquo International Journalof Distributed Sensor Networks vol 2010 Article ID 671385 17pages 2010

[4] W Chen and N Huang ldquoStrongly connecting problem onmultihop packet radio networksrdquo IEEE Transactions on Com-munications vol 37 no 3 pp 293ndash295 1989

[5] C C Shen and Z Huang ldquoTopology control for Ad Hoc net-works present solutions and open issuesrdquo inHandbook onTheo-retical and Algorithmic Aspects of Sensor Ad Hoc Wireless andPeer-to-Peer Networks J Wu Ed CRC Press New York NYUSA 2005

[6] W Song X Li O Frieder and W Wang ldquoLocalized topologycontrol for unicast and broadcast in wireless ad hoc networksrdquoIEEE Transactions on Parallel and Distributed Systems vol 17no 4 pp 321ndash334 2006

[7] M Y Aalsalem J Taheri and A Y Zomaya ldquoA framework forreal time communication in sensor networksrdquo in Proceedingsof the 2010 ACSIEEE International Conference on ComputerSystems and Applications (AICCSA rsquo10) pp 1ndash7 May 2010

[8] T Yang Y Sun J Taheri and A Y Zomaya ldquoDLS a dynamiclocal stitching mechanism to rectify transmitting path frag-ments in wireless sensor networksrdquo Journal of Network andComputer Applications vol 36 no 1 pp 306ndash315 2013

[9] V Rodoplu and T H Meng ldquoMinimum energy mobile wirelessnetworksrdquo IEEE Journal on Selected Areas in Communicationsvol 17 no 8 pp 1333ndash1344 1999

[10] W Ye J Heidemann and D Estrin ldquoMedium access controlwith coordinated adaptive sleeping for wireless sensor net-worksrdquo IEEEACM Transactions on Networking vol 12 no 3pp 493ndash506 2004

[11] T van Dam and K Langendoen ldquoAn adaptive energy-efficientMAC protocol for wireless sensor networksrdquo in Proceedings ofthe International Conference on Embedded Networked SensorSystem pp 171ndash180 November 2003

[12] G Lu B Krishnamachari and C S Raghavendra ldquoAn adaptiveenergy-efficient and low-latency MAC for data gathering in

wireless sensor networksrdquo in Proceedings of the 18th Interna-tional Parallel and Distributed Processing Symposium (IPDPSrsquo04) pp 3091ndash3098 April 2004

[13] W B Heinzelman A P Chandrakasan and H Balakrish-nan ldquoAn application-specific protocol architecture for wirelessmicrosensor networksrdquo IEEE Transactions onWireless Commu-nications vol 1 no 4 pp 660ndash670 2002

[14] S Lindsey and C S Raghavendra ldquoPEGASIS power-efficientgathering in sensor information systemsrdquo in Proceedings of theIEEE Aerospace Conference vol 3 pp 1125ndash1130 2002

[15] Y Ting and K ChunJian ldquoAn energy-efficient and fault-tolerant convergecast protocol in wireless sensor networksrdquoInternational Journal of Distributed Sensor Networks vol 2012Article ID 429719 8 pages 2012

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an MST-based topology control algorithmrdquo in Proceedings of the 22ndAnnual Joint Conference on the IEEE Computer and Communi-cations Societies (IEEE INFOCOM rsquo03) vol 3 pp 1702ndash1712April 2003

[17] X Li Y Wang and W Song ldquoApplications of 120581-local MSTfor topology control and broadcasting in wireless Ad Hoc net-worksrdquo IEEE Transactions on Parallel and Distributed Systemsvol 15 no 12 pp 1057ndash1069 2004

[18] M X Cheng M Cardei J Sun et al ldquoTopology control of adhoc wireless networks for energy efficiencyrdquo IEEE Transactionson Computers vol 53 no 12 pp 1629ndash1635 2004

[19] A Konstantinidis ZQingfu Y Kun andH Ian ldquoEnergy-awaretopology control in sensor networks using modern heuristicsrdquoin Proceedings of the Global Telecommunications Conference(IEEE GLOBECOM rsquo06) pp 1ndash5 December 2006

[20] W Guo H Gao G Chen H Cheng and L Yu ldquoA PSO-basedtopology control algorithm inwireless sensor networksrdquo in Pro-ceedings of the 5th International Conference onWireless Commu-nications Networking and Mobile Computing (WiCOM rsquo09) pp3406ndash3409 September 2009

[21] L F Liu and Y Liu ldquoTopology control scheme based onsimulated annealing algorithm in wireless sensor networksrdquoTongxin XuebaoJournal on Communications vol 27 no 9 pp71ndash77 2006

[22] R Montemanni L M Gambardella and A K Das ldquoThe mini-mum power broadcast problem in wireless networks a simu-lated annealing approachrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo05) vol4 pp 2057ndash2062 March 2005

[23] Z Huang and C C Shen ldquoDistributed topology control mech-anism for mobile Ad Hoc networks with swarm intelligencerdquoACM SIGMOBILE Mobile Computing and CommunicationsReview vol 7 no 3 pp 21ndash22 2003

[24] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[25] V Latora and M Marchiori ldquoEfficient behavior of small-worldnetworksrdquo Physical Review Letters vol 87 no 19 Article ID198701 pp 1ndash4 2001

[26] R Cohen K Erez D Ben-Avraham and S Havlin ldquoResilienceof the Internet to random breakdownsrdquo Physical Review Lettersvol 85 no 21 pp 4626ndash4628 2000

[27] V Latora and M Marchiori ldquoEconomic small-world behaviorin weighted networksrdquo European Physical Journal B vol 32 no2 pp 249ndash263 2003


[28] H Wang and P V Mieghem ldquoAlgebraic connectivity optimiza-tion via link additionrdquo in Proceedings of the 3rd InternationalConference on Bio-Inspired Models of Network Information andComputing Sytems (BIONETICS rsquo08) pp 1ndash8 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



There are many factors causing the dynamic structuremdashfrom system hardware to applicationmdashfor unattended sensornodes with miniature sizes (mm scale for smart dust motes)limited battery-power and low reliable hardware circuitswhen coping with harsh conditions Other factors thatmay affect network connectivity and communication amongsensor nodes are fading signal strength obstacles weatherconditions interference and so forth [7 8] An immutabletopology structure is not enough for the WSNs and anydynamic change will break original optimization and reducethe network performance

To overcome this critical problem this paper proposeda novel betweenness addition edges expansion algorithm(BAEE) With the betweenness parameter BAEE algorithmexpanded the minimum-cost edges to optimize the networktopology with maximum improving of the efficiency func-tion values The preliminary simulation results comparedwith Fiedler-vector-based strategy showed that our algorithmcould obtain more robust topology with higher invulnera-bility under both the random failures and intentional attackscenarios

This paper is organized as follows Section 1 introducesthe TOC problem in WSNs and Section 2 presents therelated work The problemrsquos mathematic description andmodel building are presented in Section 3 Section 4 proposesthe BAEE algorithm in detail Section 5 presents simulationresults to demonstrate the effectivity of the algorithm Section6 concludes the paper

2 Related Work

There are three types of approaches in the previous TOCresearch presented as follows (1) Control each nodersquos emis-sion power to reduce the strong connectivity of the physicaltopology and to effectively save the energy consumption andprolong network lifespan Rodoplu andMeng [9] introducedthe notion of relay region and enclosure for the purpose ofpower control It was shown that the network was stronglyconnected if every node maintained links with the nodesin its enclosure With reducing the transmission power thetopology connectivity becomes thin Building a minimum-power-connected topology is a multiobjective optimizationproblem (2) Reduce the total number of working nodesin WSNs and let other nodes suspend to hibernate It canalso reduce the topology complexity Moreover the approachhelps to reduce the interference that exists in wireless net-work which means that a greater signal-to-noise ratio willbe obtained at receiving nodes The most common schemesbased on this principle are sensor-MAC (S-MAC) [10]timeout-MAC (T-MAC) [11] and data-gathering MAC (D-MAC) [12] (3) Control sensor nodersquos logical degree in its log-ical topology thus helping to reduce MAC layer contentionand improve space reuse A less nodersquos logical degreemay alsohelp to mitigate the hidden and exposed terminal problemsClustering topology control strategy is one of the effectiveapproaches similar to spanning-tree structure in WSN Thelow-energy adaptive clustering hierarchy (LEACH) [13] isthe most notable clustering algorithm for wireless sensornetworks LEACH combines the ideas of energy-efficient

cluster with application-specific data aggregation to achievegood performance Its improved algorithm power-efficientgathering in sensor information systems (PEGASIS) [14] isa chain-based clustering scheme Another effective topologystructure is the spanning-tree [15] Li et al [16] proposeda fully distributed topology control algorithm called LMSTA similar method 119896-local MST was addressed by Li et al[17]

With the number of sensor nodes increasing the topol-ogy of large-scale WSNs becomes more and more complexand TOC as a type of multiobjective optimization problemsis very difficult to explore the global optimal solution suchas the degree-constrained minimum spanning-tree problemSome heuristic methods were developed to improve the opti-mization performance [18ndash23] In [18] the authors proposedtwo heuristics based on a minimum spanning-tree algorithmand a broadcast incremental power method respectivelyKonstantinidis developed a genetic algorithm with localsearch that performs better than the MST heuristic [19] Guopresented an improved discrete particle swarm optimizationalgorithm for generating topology schemes [20] A simulatedannealing algorithm was designed in [21] and it is alsoapplied to solve the problem of minimizing broadcast treeone type of the physical topology control problems [22] In[23] ant colony optimization a framework inspired by the antforaging behavior in the area of swarm intelligence is appliedto physical topology control

The above heuristic algorithms focused on the solutionprocedure of optimization problem itself in which topologycontrol had been abstracted into the multiobjective opti-mization problem On the different view to analyze thetopology control problem we use the complex networktheory to calculate the networkrsquos long-range and short-rangeconnectivity and then a novel BAEE algorithm is proposedto improve the networksrsquo robustness and invulnerability withthe minimum-cost edges expanded

3 The Network Model and the Parameters ofComplex Network

The formal definition of the TOC problem in WSNs ispresented as follows In a special sensor area there are a setof 119899 wireless nodes 119881 = V

1 V2 V

119899 119864 = 119890

1 1198902 119890

119899

is the set of communication links When an adjacent pair V119895

V119896shows the same wireless medium 119890

119894(V119895 V119896) indicates that

both nodes are within their wireless transmitting ranges 1205820

that is 119864 = 119890119894(V119895 V119896) | 119863(V

119895 V119896) le 1205820 V119895 V119896isin 119881 Therefore

the wireless sensor network is represented as a simple digraph119866 = (119881 119864) Because of the omnidirectional antenna inWSNsany two nodes can communicate if the Euclidean distancebetween them is less than the communication range There-foreWSNsrsquo topology is usually strongly connectedThe com-plex network theory is utilized to analyze this type of stronglyconnected topology in this paper Some parameters of com-plex network are presented firstly in the following section

31 The Parameters of Complex Network A complex net-workrsquos attribute can be described by its key parameters degree








119875119888(119896) =

119896max

sum

1198961015840ge119896

119901 (1198961015840) (1)










119862119894=

120582119866(V)

119879119866(V)

(2)


119862 =1

119873

119873

sum

119894=1

119862119894 (3)



119871 =1

119873 (119873 minus 1) 2sum

1le119894119895le119873

119889119894119895 (4)




119861 (V) = sum

V = 119894 V = 119895 119894 = 119895

120590119894119895(V)

120590119894119895

(5)



pass through V



119877 (119866) =

119878 (1198661015840)

119873 minus 119873119903

(6)




119903nodes

were removed




= 1119889119894119895



119894119895= 0The global


119864glob (119866) =

sum119894 = 119895isin119866

120576119894119895

119873(119873 minus 1)=

1

119873 (119873 minus 1)sum

119894 = 119895isin119866

1

119889119894119895

(7)


119864loc (119866) =1

119873sum

119894isin119866

119864 (119866119894)

119864loc (119866) =1

119896119894(119896119894minus 1)

sum

119897 =119898isin119866

1

1198891015840

119897119898

(8)









119894119895= 1 forall119894 119895




119886119894119895

= 1 119890119894119895

isin 119864

0 otherwise(9)







119894or edge


119894and 119890

119894in network










119861119899= [119887 (V

1) 119887 (V

2) 119887 (V

119899)]119879

(10)



]119898is calcu-

lated where 119898 = 1198622

119899 as follows

119861+

(119896) = 119887 (V119894) + 119887 (V

119895)

119896 = (119894 minus 1) times 119899 + 119895

(11)


+

]1015840

119898




1015840

119898have







(4) Check [119861+

]1015840


+(0) For

119861+

(119896) = 119887(V119894) + 119887(V


119894and V

119895have



119898















1015840

BAEE and 1198661015840





1015840



1015840



1015840





1015840


1015840

BAEE) and119877(1198661015840



1015840


1015840


1015840


1015840

BAEE) and 119864(1198661015840




1015840




1015840

BAEE)

value than FVBS 119864(1198661015840





119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895


1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)



(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895



119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)





BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)




6 Conclusions


0010203040506070809

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)




0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119875119888(119896) =

119896max

sum

1198961015840ge119896

119901 (1198961015840) (1)










119862119894=

120582119866(V)

119879119866(V)

(2)


119862 =1

119873

119873

sum

119894=1

119862119894 (3)



119871 =1

119873 (119873 minus 1) 2sum

1le119894119895le119873

119889119894119895 (4)




119861 (V) = sum

V = 119894 V = 119895 119894 = 119895

120590119894119895(V)

120590119894119895

(5)



pass through V



119877 (119866) =

119878 (1198661015840)

119873 minus 119873119903

(6)




119903nodes

were removed




= 1119889119894119895



119894119895= 0The global


119864glob (119866) =

sum119894 = 119895isin119866

120576119894119895

119873(119873 minus 1)=

1

119873 (119873 minus 1)sum

119894 = 119895isin119866

1

119889119894119895

(7)


119864loc (119866) =1

119873sum

119894isin119866

119864 (119866119894)

119864loc (119866) =1

119896119894(119896119894minus 1)

sum

119897 =119898isin119866

1

1198891015840

119897119898

(8)









119894119895= 1 forall119894 119895




119886119894119895

= 1 119890119894119895

isin 119864

0 otherwise(9)







119894or edge


119894and 119890

119894in network










119861119899= [119887 (V

1) 119887 (V

2) 119887 (V

119899)]119879

(10)



]119898is calcu-

lated where 119898 = 1198622

119899 as follows

119861+

(119896) = 119887 (V119894) + 119887 (V

119895)

119896 = (119894 minus 1) times 119899 + 119895

(11)


+

]1015840

119898




1015840

119898have







(4) Check [119861+

]1015840


+(0) For

119861+

(119896) = 119887(V119894) + 119887(V


119894and V

119895have



119898















1015840

BAEE and 1198661015840





1015840



1015840



1015840





1015840


1015840

BAEE) and119877(1198661015840



1015840


1015840


1015840


1015840

BAEE) and 119864(1198661015840




1015840




1015840

BAEE)

value than FVBS 119864(1198661015840





119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895


1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)



(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895



119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)





BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)




6 Conclusions


0010203040506070809

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)




0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119894119895= 1 forall119894 119895




119886119894119895

= 1 119890119894119895

isin 119864

0 otherwise(9)







119894or edge


119894and 119890

119894in network










119861119899= [119887 (V

1) 119887 (V

2) 119887 (V

119899)]119879

(10)



]119898is calcu-

lated where 119898 = 1198622

119899 as follows

119861+

(119896) = 119887 (V119894) + 119887 (V

119895)

119896 = (119894 minus 1) times 119899 + 119895

(11)


+

]1015840

119898




1015840

119898have







(4) Check [119861+

]1015840


+(0) For

119861+

(119896) = 119887(V119894) + 119887(V


119894and V

119895have



119898















1015840

BAEE and 1198661015840





1015840



1015840



1015840





1015840


1015840

BAEE) and119877(1198661015840



1015840


1015840


1015840


1015840

BAEE) and 119864(1198661015840




1015840




1015840

BAEE)

value than FVBS 119864(1198661015840





119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895


1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)



(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895



119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)





BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)




6 Conclusions


0010203040506070809

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)




0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















1015840

BAEE and 1198661015840





1015840



1015840



1015840





1015840


1015840

BAEE) and119877(1198661015840



1015840


1015840


1015840


1015840

BAEE) and 119864(1198661015840




1015840




1015840

BAEE)

value than FVBS 119864(1198661015840





119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895


1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)



(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895



119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)





BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)




6 Conclusions


0010203040506070809

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)




0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119861(V) = sum

V = 119894V = 119895119894 = 119895

120590119894119895(V)

120590119894119895


1) 119887(V2) 119887(V

119899)]119879

(3) For 119894 = 0 to n(4) For 119895 = 119894 + 1 to 119899

(5) 119896 = (119894 minus 1) times 119899 + 119895 119861+(119896) = 119887(V119894) + 119887(V

119895)



(10) If 119861+

119894lt 119861+

119895

(11) Switch 119861+

119894and 119861

+

119895



119895 that is 119890(V

119894 V119895) notin 119864 here 119861

+

(119896) = 119887(V119894) + 119887(V

119895)





BAEE and 1198661015840

FVBS

1198661015840

BAEE (61 79) (80 34) (45 78) (51 2) (54 73) (50 57) (55 22) (56 68)1198661015840

FVBS (100 1) (100 2) (99 1) (100 3) (99 2) (100 4) (99 3) (100 5)




6 Conclusions


0010203040506070809

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)




0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



0010203040506070809

1


Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)







0

005

01

015

02

025

03


Effici

ency

func

tionE(G)



07

075

08

085

09

095

1



Con

nect

ivity

robu

stnes

s fun

ctio

nR(G)


01012014016018

02022024026


Effici

ency

func

tionE(G)





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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